01 - Preheat oven to 350°F. Grease a quarter sheet pan (9x13 inches) and line with parchment paper.
02 - Whisk together cake flour, baking powder, baking soda, and salt in a medium bowl.
03 - Using a stand mixer, beat butter, shortening, granulated sugar, and brown sugar on medium-high speed until light and fluffy, about 2-3 minutes. Scrape down the bowl.
04 - Add eggs one at a time, beating well after each addition. Reduce speed to low and add buttermilk, whole milk, and vanilla extract. Mix until combined.
05 - Gradually add dry ingredients on low speed, mixing just until incorporated. Fold in rainbow sprinkles by hand.
06 - Pour batter into prepared pan and smooth the top. Bake for 28-32 minutes until a toothpick inserted in the center comes out clean. Cool completely in the pan.
07 - Lower oven temperature to 300°F. In a bowl, combine flour, granulated sugar, brown sugar, baking powder, salt, and sprinkles. Add melted butter and vanilla, mixing with a fork until small clusters form.
08 - Spread crumb mixture on a lined baking sheet. Bake for 15 minutes, stirring halfway through. Cool completely.
09 - Beat butter, shortening, and cream cheese until smooth. Add vanilla and salt. Gradually beat in powdered sugar until fluffy. Add milk as needed for spreadable consistency.
10 - Whisk together whole milk and vanilla extract in a small bowl.
11 - Using a 6-inch cake ring, cut two full circles and two half circles from the cooled cake sheet.
12 - Place a cake round inside a lined 6-inch cake ring set on a cake board. Brush with vanilla soak. Spread 1/5 of frosting over the cake, sprinkle 1/3 of the crumbs. Repeat layering with half-circles, soak, frosting, and crumbs.
13 - Top with final cake round, soak, and finish with remaining frosting and crumbs. Chill for at least 1 hour or overnight. Remove ring and serve at room temperature.